LONDON — For decades British leaders have relied on meetings with American presidents to show off their influence over a global superpower and illustrate the durability of the “special relationship” they prize so much.

But for Prime Minister Theresa May, who has endured months of political turbulence, including the recent resignation of Boris Johnson, her foreign secretary, President Trump’s impending trip to Britain looks like it could be an ordeal more than anything else.

Even before his arrival, Mr. Trump stirred the pot, suggesting that he would like to catch up with his “friend” Mr. Johnson while in Britain, a country that the president described as “in somewhat turmoil.”

When asked whether Mrs. May should stay in her job, Mr. Trump adopted the pose of a disinterested observer, pronouncing this a question for “the people.” He then suggested that speaking with President Vladimir V. Putin of Russia, which he is scheduled to do next week, might be easier than conversing with Mrs. May, who will host a dinner and a lunch with the president during his stay.

Yet the prime minister has little choice but to swallow her pride and get on with it, as she herself likes to say, because she wants Mr. Trump’s help for a trade deal with the United States after Britain’s withdrawal from the European Union, known as Brexit. And, anyway, compared with wrestling with her government, which this week suffered three ministerial resignations, spending a couple of awkward days with Mr. Trump might actually be a relief.

Beyond the psychodramas and the expected protests at every stop on Mr. Trump’s itinerary, the visit also raises more profound questions about the durability of the special ties between London and Washington at a time when Mr. Trump is attacking the basic institutions of the postwar international order and with Britain on the verge of leaving the European Union.

No one argues that the relationship between Mr. Trump and Mrs. May is anything like that between such venerated predecessors as Winston Churchill and Franklin D. Roosevelt, or akin to the political romance of Ronald Reagan and Margaret Thatcher. But is it a sign of bad personal chemistry or of something deeper, a widening divide that might never be sewn back up?

Optimists in Britain say the deep economic, military, intelligence-sharing and cultural ties across the Atlantic are strong enough to survive the occasional Trump Twitter storm.

“The ‘special relationship’ is much deeper than one man and one woman in office, it is a multilayered partnership,” said Tom Tugendhat, chairman of the Foreign Affairs Select Committee in the House of Commons.

“It’s not ending,” added Leslie Vinjamuri, head of the United States and Americas program at Chatham House, a research institute, speaking of the deep bond. “People are looking in the wrong place.”

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In 1988, Prime Minister Margaret Thatcher of Britain danced with President Ronald Reagan at a state dinner given in her honor at the White House.CreditM. Sprague/Agence France-Presse — Getty Images

“The ‘special relationship’ is much bigger than two individuals — it’s about history, culture, institutions, it’s about shared interests, that wax and wane over time,” she said. “On this particular visit, people are going to read into the ‘special relationship’ based on the charisma that does or doesn’t exist. But we’ve had downturns in the past.”

Others are not so confident. Nigel Sheinwald, a former British ambassador to the United States, said that there was a clear contrast with his time in Washington and when he worked as a foreign policy adviser to former Prime Minister Tony Blair, with whom President George W. Bush would talk of “co-strategizing.”

In fact, Mr. Sheinwald said, the trans-Atlantic relationship has been of declining importance to the United States for most of its 70-year life, though this was sometimes masked by the warmth of personal ties between individual presidents and prime ministers.

The difference now is that the “special relationship” is “part of the post-World War II political and economic system, and that order is under threat,” he said. “The combination of President Trump’s administration, his election, his policies and Brexit, all combined, represent quite a challenge for the U.S.-U.K relationship.”

After Brexit, Britain is banking on a future outside the European Union, based on free trade and policed by the World Trade Organization. Yet that is one of the international bodies that Mr. Trump sees as part of the problem as he embarks on a trade war that has embroiled European nations.

The context of the visit illustrates vividly the wider strategic challenge. Mr. Trump arrived on Thursday from a tense meeting of NATO leaders, officially his closest allies, in Brussels. On Sunday, he will head to Helsinki, Finland, for his meeting with Mr. Putin.

Mr. Trump “has different rules and they are very destabilizing for other politicians who expect an alliance leader to be aware of the fragility of politics in their countries and to do something to help them,” Mr. Sheinwald added.

Even if there is little sign of such help, Britain’s government will do its best to roll out the red carpet, playing to Mr. Trump’s appetite for flattery and hoping this will at least give him a warm feeling and greater sense of connection to the British.

There will be a black-tie dinner at Blenheim Palace — Churchill’s birthplace — a meeting with Mrs. May at her 16th-century country residence, Chequers, and tea with Queen Elizabeth II at Windsor Castle. Then it will be on to Scotland, where Mr. Trump’s mother was born.

But British officials fear that if the NATO summit meeting finishes anything like as badly as the recent Group of 7 meeting — which ended with biting criticism of Prime Minister Justin Trudeau of Canada by Mr. Trump — it will inevitably overshadow the visit to Britain.

One drawback for Mrs. May is that she needs more from Mr. Trump than he needs from her and that leaves her in a weak position — and Mr. Trump has made no secret of his contempt for weakness.

Certainly, she cannot rely on personal rapport. The daughter of a clergyman, Mrs. May has a cautious, provincial approach to politics that is the exact opposite of Mr. Trump’s brash showmanship and fondness for breaking the diplomatic furniture.

She seemed to have scored a coup by making an early visit to Washington in January 2017 and promising Mr. Trump a full state visit. That offer was quietly pushed back because of opposition in Britain, and trans-Atlantic relations cooled further when Mrs. May criticized Mr. Trump after he retweeted a video from a far-right British group and again for his policy of separating children from their refugee parents.

As many analysts have pointed out, no matter what strategy European leaders have tried in dealing with Mr. Trump — distance from Chancellor Angela Merkel of Germany, charm from France’s president, Emmanuel Macron — none have worked.

That is because, whatever the body language, there is only limited common ground on policy. High on Mrs. May’s agenda will be discussions of a trade deal that might help redress some of the likely economic hit Britain faces from Brexit.

Mr. Trump, an early Brexit supporter, could be helpful, analysts say.

“If the U.K. gets out of Europe, and Trump gets more angry at Germany, I wouldn’t be surprised if he cuts a deal with the U.K. bilaterally,” said Ms. Vinjamuri, who added that Mr. Trump would prefer to deal with a more isolated Britain that he could dominate.

“Teams give him the sense that they gang up on him and they’re something he can’t control,” she said. “If the U.K. makes this move that will be more interesting.”

Yet, even here there are minefields. That feeling of isolation is not what Mrs. May wants. Her plan is for Britain to remain closely aligned to European standards on goods and agricultural products, even if that inevitably leaves less scope for an ambitious trade deal with the United States.

But Mr. Trump’s sympathies lie with those like Mr. Johnson and Nigel Farage, former leader of the pro-Brexit U.K. Independence Party, who are pressing for a cleaner break with the European Union.

Both Mr. Johnson and Mr. Farage have accused Mrs. May of weakening the terms of Britain’s exit and, if Mr. Trump makes it clear that he agrees with them, that would damage her.

Meanwhile, on big international questions, such as climate change, Iran and the Middle East, Mrs. May is in the awkward position of being much closer to a European Union that she wants to quit than to Mr. Trump.

In recent years, Britain has tried to act as a trans-Atlantic bridge between Washington and Brussels. It will lose that role once it is outside the bloc, and there are rivals for Mr. Trump’s attention. It is, for example, a year since he visited Paris as the guest of Mr. Macron in France, where his trip unfolded without significant public protests.

That probably means that to win American’s favor in Mr. Trump’s more Darwinian diplomatic world, British leaders will either have to spend more on the military, security and other tools of international projection or side with the United States on foreign policy issues, perhaps including the Middle East, that may be deeply unpopular at home.

“The government would like it to be otherwise, but I can’t see anything other than Brexit diminishing the U.K. internationally,” Mr. Sheinwald said.

“It limits the connectivity that was the key to the U.K’s importance as an international player,” he added. “If we are not at the E.U’s table we are going to be less valuable allies for countries like America.”